Suppose that U is a 3-by-3 unitary matrix. Prove that U is a normal matrix and prove that there exists an orthonormal basis u1, U2, U3 of C° and complex numbers C1, C2, C3 with U = c¡u¡u* + c2u2u" + c3U3u such that |c|| = |c2| = |c3| = 1.

Let B = {v1, V2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B. • (a) Is (,) an inner product? (b) Can you find a matrix A (constructed using v1, V2, and v3) such that (u, v) = u" Av?

Let {v1, v2, . .. , vn} be an orthogonal set in R" with respect to the Euclidean inner product. Let A be the n x n matrix with row vectors v1, v2, ... , Vn. b) What extra condition(s) do we need from the vectors v1, v2, ..., Vn so that the matrix AAT is an identity matrix? Justify your answer!

6) PLEASE ANSWER EACH QUESTION, THANKS.